Let V be a vector space over a division ring D. A subring R of Hom_D(V,V) is said to be n-fold transitive if for every k (1 ≤ k ≤ n) and every linearly independent subset I u₁, . .. , u_k} of V and every arbitrary subset {v₁, ... , v_k} of V, there exists θ ϵ R such that θ(u_i) = v_i for i = 1,2, ... , k.

(a) If R is one-fold transitive, then R is primitive. 

(b) If R is two-fold transitive, then R is dense in Hom_D(V,V).